A simple harmonic oscillator of amplitude A has a total energy E. Determine a) the kinetic energy and b) the potential energy when the position is one third the amplitude. c) For what values of the position does the kinetic energy equal one half the potential energy

a) Potential energy:

b) Kinetic energy:

c)

Explanation:

a)

The total energy of a simple harmonic oscillator is given by:

where

k is the spring constant

A is the amplitude (the maximum displacement) of the system

The potential energy can be written as

where

x is the displacement

Here, the position is 1/3 of the amplitude, so

This means that the potential energy is:

b)

The total energy is conserved during the motion, and therefore, at any instant, it is sum of the kinetic energy and the potential energy of the system:

(1)

where

K is the kinetic energy

U is the potential energy

As we found in part a), the potential energy when the position is 1/3 of the amplitude is

(2)

Therefore, substituting (2) into (1) and solving for K, we find the kinetic energy of the system:

c)

Here we want to find the value of the position at which the kinetic energy equal one half the potential energy.

Mathematically, this means that

Using again the formula of the total mechanical energy

And substituting K, we find

(2)

This means that the kinetic energy is one half the potential energy when the potential energy is 2/3 of the total energy.

Using the expression for the total energy:

And the expression for the potential energy:

And combinng them into (2), we find:

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